arXiv subjects

Goran Radić

Publications and source records attributed to Goran Radić.

2 recordsLinked to original sources

arXiv2026

Structural aspects of $1$-limited domination in cubic graphs

A set $D\subseteq V(G)$ is a $1$-limited dominating set of a graph $G$ if every vertex outside $D$ has a neighbor in $D$ and every vertex of $D$ has at most one neighbor outside $D$. The minimum cardinality among all $1$-limited dominating sets of $G$ is denoted by $γ_1^{\mathrm L}(G)$. For every cubic graph $G$ of order $n$, there exists a nonnegative integer $t$ such that $γ_1^{\mathrm L}(G)=\frac n2+t$. We establish structural properties of $1$-limited dominating sets in terms of $t$ and use them to study the induced cycles vertex number, that is, the maximum number of vertices inducing a $2$-regular subgraph. As applications, we determine the exact $1$-limited domination numbers of generalized Blanuša snarks and flower snarks.

math.CO↗

arXiv2026

On $1$-limited and $(1,2)$-domination in cubic graphs

A dominating set $D$ of a graph is called $1$-limited if every vertex of $D$ has at most one neighbor outside $D$, while a $(1,2)$-dominating set is a dominating set in which every vertex of the set has at least two neighbors within the set. These two notions coincide on cubic graphs. We prove that the decision problem 1-Limited Dominating Set is $\mathsf{NP}$-complete even when restricted to $2$-connected planar cubic graphs, thereby completing the known complexity results for $k$-Limited Dominating Set for all fixed positive integers $k$. We also determine the exact $1$-limited domination number of the entire Goldberg family. This provides a further infinite family of cubic graphs supporting several open conjectures and proposed bounds concerning $(1,2)$-domination and induced cycles.

math.CO↗